1. Cognitive Radio Technologies, LLC.

2. Wireless @ Virginia Tech, Bradley Department of Electrical and Computer Engineering, Virginia Tech.

3. Mobile and Portable Radio Research Group, Virginia Tech

4. Department of Economics, Virginia Tech.

This material is based in part on work sponsored by basic research grant no. N000140310629 from the Office of

Naval Research, the IREAN program at Virginia Tech, NSF CAREER grant no. CNS-0448131, and the support of

MPRG and Wireless @ Virginia Tech industrial affiliates.

A Low Complexity Dynamic Frequency Selection Algorithm

for Cognitive Radio Networks

James O. Neel1, Member IEEE, Jeffrey H. Reed

1,2,3, IEEE Fellow, Robert P. Gilles

4, Allen B.

MacKenzie2, Member IEEE

Abstract

While spectrum filling via dynamic frequency selection (DFS) is a commonly proposed

application for cognitive radio (CR), implementations that account for interactive effects are

traditionally of high complexity or involve significant signaling overhead. By applying the

controlled observation option of the CR design framework we proposed in [1] based on the

theory of potential games, we developed a low complexity distributed autonomous ad-hoc

dynamic frequency selection algorithm that can be provably shown to converge to near-minimal

interference frequency re-use patterns. Repeated random trials of this algorithm show an average

reduction in net interference levels of 19 dB.

Index Terms: dynamic frequency selection, cognitive radio, game theory, potential games,

interference reducing networks, bilateral symmetric interference

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I INTRODUCTION

Consider a system of three wireless links {1,2,3} which connect three access nodes (AN) with

three clients. To minimize the amount of interference experienced by its clients on the downlink,

each As switches between two orthogonal channels, {0,1}. Suppose g31>g21, g12>g32, g23>g13

where gij is the gain from the AN of link i to the client of link j. In other words, client 1 is

interfered with more by AN 3 than by AN 1; client 2 is interfered with more by AN 1 than by

AN 2; and client 3 is interfered with more by AN 2 than by AN 1. Without a loss of generality,

assume g31 = g12 = g23 = 1.0 and g21 = g32 = g13 = 0.5 and a transmit power of 1 so the observed

interference at a client is equal to the sum of link gains over all inband links.

For the eight possible combinations of choices, the interference levels experienced by each

client is shown in Table 1 where the first entry on the channel row specifies the choice of

channels by each AN (1,2,3) and the second row specifies the interference levels seen by the

client (1,2,3). Assuming each AN chooses the channel with the least amount of interference at its

client, the system will enter into an infinite loop – (1,0,0), (1,0,1), (0,0,1), (0,1,1), (0,1,0), (1,1,0),

(1,0,0)… – from any initial frequency allocation. So with each AN adapting to reduce its client’s

interference, the resulting network-wide behavior is undesirable as the infinite loop implies

significant bandwidth will be consumed in signaling the adaptations.

Table 1: Interference Levels for Example DFS Algorithms

Channel (0,0,0) (0,0,1) (0,1,0) (0,1,1) (1,0,0) (1,0,1) (1,1,0) (1,1,1)

Interf. (1.5,1.5,1.5) (0.5,1,0) (1,0,0.5) (0,0.5,1) (0,0.5,1) (1,0,0.5) (0.5,1,0) (1.5,1.5,1.5)

Unfortunately, the pattern of link gains which leads to this infinite recursion is not a special

case. In fact, one out of every four deployments of this system will enter into an infinite loop!1 In

fact, as the number of coexisting links grows, the probability of the network entering an infinite

1 Note that the set of relationships g31>g21, g12>g32, g23>g13 occurs with probability 1/8 for randomly assigned gains.

Then note that a loop in the opposite direction would occur with probability 1/8.

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loop rapidly approaches 1.2 Even increasing the number of channels does not eliminate this

problem as long as the number of channels is less than the number of adapting links – a situation

that seems guaranteed for practical wireless systems. This example illustrates a critical challenge

to deploying CRs –multiple adapting CRs spawn interactive decision processes whose outcomes

are influenced by the adaptations of all of the links, not just the adaptation of a single link which

means network-wide behavior must be accounted for in the design of CR algorithms.

To analyze the network-wide behavior of interactive CRs, several authors have proposed

modeling CR networks as a game and analyzing the system with game theory. In this paper, we

use game theory to specify a combination of observations and goals which ensure the selfish

frequency adaptations of CRs never enter an infinite loop. As we showed in [1], our approach

ensures loop-free convergence to a frequency allocation which is a minimizer of sum network

interference. In this paper, we characterize the performance of this algorithm and extend the

algorithm to ad-hoc networks. The remainder of this paper is organized as follows. Section II

presents prior work related to cognitive DFS. Section III describes the theoretical framework

used to design these algorithms. Section IV shows how this framework can be applied to the

design of DFS algorithms for infrastructure-based networks. Section V extends these results to

ad-hoc networks and Section VI characterizes the steady-state behavior of the network.

II RELATED WORK

Many authors have attacked the infinite loop by requiring explicit coordination between nodes or

by assuming a centralized decision maker. After noting that finding the optimal frequency

allocation is a NP-complete problem, [2] proposes a heuristic centralized algorithm based on a

local search algorithm with random restart to search through the possible frequency

combinations considering only the interactions of the access points. As part of a solution to

network formation problem [3] utilizes a central controller to assign frequencies to each link in

2 The probability of an infinite loop can be bounded as the probability of any three links having the looping gain

relationship, or for n links [ ] ( ) 31 3/ 4 n CP loop ≥ − for n≥3.

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the network according to the abbreviated algorithm summarized as follows. First the controller

determines the frequencies each device can use and not cause interference to higher priority

devices. Then the controller determines the frequencies for each device which do not cause

interference to the neighbors of higher priority devices. Then the algorithm picks a node and over

all of its connections assigns a non-interfering frequency to that edge. This continues until all

links are assigned a frequency or the algorithm fails.

Other authors do not assume a central controller, but instead assume extensive message

passing between the devices so each radio can effectively calculate the same solution. For

instance, [4] considers a network of orthogonal channels where adaptive secondary users

coordinate their adaptations via a common channel. [5] considers a system wherein optimal

frequency/power allocations are achieved by applying punishment strategies to DFS when the

optimal strategy is known a priori. [6] considers a Dynamic Spectrum Access (DSA) scheme

wherein radios must share information over a common channel to compute the interference

levels each radio would induce to other radios in order to evaluate its goal, which is the sum of

every device’s observed interference. While this has the virtue of being both an exact potential

game and an IRN (specifically, a globally altruistic IRN), it requires significant overhead to

distribute the information needed to evaluate the goal and the authors require the decisions be

made sequentially. For DSA systems where spreading codes are adapted3, [7] presents an

algorithm where each radio’s goal incorporates the interference measurements of all other radios

in the system (another globally altruistic IRN). [8] considers a Homo Egualis (“fair man”)

implementation where each access point chooses frequencies so as to maximize (1) where xj is

the usable spectrum for user j and αi,βi∈�. Thus each access point attempts to ensure that every

access point is receiving approximately the same amount of spectrum.

3 In the context of signal space representations, spreading code adaptation algorithms could be directly applied to

DFS problems. However, proper synchronization may be a practical constraint.

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( ) ( )1 1

j i j i

i ii i j i i j

x x x x

u x x x x xn n

α β

> <

= − − − −− −∑ ∑ (1)

[9] considers a distributed graph coloring algorithm where edges are formed between interfering

access points. Each AN recursively broadcasts frequency and interference measurements and

selects the frequency it believes will result in the least interference. Other authors have

considered single cell adaptations without the need for communication beyond reporting

measurements from a common receiver. [10], [11], [12], and [13] consider spreading code

adaptations where each AN is isolated in frequency and spreading codes are chosen to minimize

the interference of clients/mobiles.

[6] also proposes another goal (or utility function) for DSA (U1) that is identical to the goal

used in this paper. However, because [6] places no restrictions on the radio’s observation

mechanism, [6] is unable to show that their system forms an exact potential game which would

permit the use of a simple distributed and autonomous algorithm. Instead [6] employs a no-regret

learning algorithm wherein the radios autonomously try every possible frequency and then adapt

to frequencies that yield the best weighted cumulative utility and show that the algorithm

converges to a mixed-strategy equilibrium – a less than optimal result as mixed strategies in

frequency selection imply continuous probabilistic adaptation which is functionally equivalent to

an infinite loop. [14] considers a related algorithm applied to a regular 10x10 grid of access

points where each radio is guided by (2) where Mi is the number of users attached to AN i, Sk is

the set of nodes operating on fk and f evaluates the throughput for the number of users in the

argument. Each AN then chooses the channel that maximizes its throughput and switches to it

with a fixed probability, p.

( )j k

j k

ii k j

M Sj

M S

Mu f f M

M ∈

∈

=

∑∑

(2)

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By applying the interference reducing network (IRN) design framework of [1], this paper

proposes low-complexity autonomous distributed DFS algorithms suitable for use in

infrastructure or ad-hoc 802.11h networks where each link does not have to infer or know what

other links are experiencing. Further, these algorithms operate without looping and converge to a

minimizer of sum observed interference.

III A GAME THEORETIC DESIGN FRAMEWORK

As has been previously described [15], a game can be used to model a CR network by

representing the CRs as players, CR adaptations as action sets, and the CR’s goal as the player’s

utility function. These modeling features are sufficient for capturing the behavior of a single

iteration of a distributed cognitive DFS algorithm in the normal form game model Γ=⟨N,F,{ui}⟩

where N denotes the set of players (radios) of cardinality n and i∈N specifies a particular player,

F represents the frequency space formed as 1 nF F F= × ×� where Fi specifies the set of

frequencies available to player i, {ui} is the set of utility functions, :iu F →� that describe the

values which the radios assign to points in F. For notational convenience, we write f to refer to a

frequency vector wherein each player in the game has chosen an frequency, fi to refer to the

frequency chosen by player i, and f-i to refer to the vector formed by considering the frequency

choices by all players other than player i.

To model multi-iteration issues such as the existence of infinite adaptation loops, the normal

form game model can be extended by incorporating the decision rules, :i id F F→ , that guide

radios’ adaptations and the decision timings, T, at which the decisions are implemented to form

the tuple, Γ=⟨N,F,{ui},{di},T⟩ [16]. In general we assume that the radios act in their own interest

in a manner which we term autonomously rational, i.e., ( )1 ,i i i if d f f−∈ , 1

i if f≠ ⇒

( ) ( )1, ,i i i i i i

u f f u f f− −> . The system behavior that results when all decision rules conform to the

autonomously rational assumption is called a better response dynamic and this is sometimes

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restricted to an exhaustive better response dynamic when all decision rules satisfy

( ),i i i if d f f−∉ if ( ) ( ): , ,

i i i i i i i if F u f f u f f− −∃ ∈ >� � .

In the infinite DFS loop example, each radio implements an autonomously rational decision

rule which can be further characterized as an exhaustive better response dynamic. The infinite

loop example also serves to illustrate that autonomous rationality is not sufficient to guarantee

that a system has a steady-state. In game theory, the traditional “steady-state” concept is the

Nash equilibrium (NE) which in this context is a frequency vector *f such that

( ) ( )* *,i i i i

u f u f f−≥ ,i i

i N f F∀ ∈ ∈ . As we showed in Theorems 1 and 2 in [1], the NE of the

normal form game model of a CR network are fixed points of 1 :n

d d F F× × →� for

autonomously rational decision rules and are the only fixed points for exhaustive better response

dynamics. Note that showing that an NE exists for the game model of a CR network and that the

network realizes an exhaustive better response dynamic does not prove the network’s steady-

states are desirable or that the network lacks infinite loops.

However, [1] presents a CR design framework which guarantees loop-free convergence to

desirable steady-states as described by the network interference function, ( ) ( )i

i N

f I f∈

Φ =∑

where :i

I F →� is the interference level observed by CR i for the frequency vectors in F. In

this design framework, a CR network, { } { }, , , ,i i

N F u d T , is said to be an interference reducing

network (IRN) if all unilateral autonomously rational adaptations decrease the value of Φ. Five

different classes of IRNs are presented in [1]: globally altruistic, locally altruistic, power

controlled isolated clusters, close proximity networks, and controlled observation networks. The

first two classes require varying degrees of coordination between the CRs and are applicable

under all operating conditions while the last three classes require no coordination between CRs

with autonomous selfish adaptations nonetheless reducing Φ but are only applicable to certain

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operating conditions. This paper applies the controlled observation networks class to design

novel low-complexity infrastructure and ad-hoc DFS algorithms.

The controlled observation networks class places a condition on the observed interference

between pairs of decision makers termed bilateral symmetric interference (BSI). In practice, BSI

holds when gjkpjρ(fj, fk) = gkjpkρ(fk, fj) ,j j k k

f F f F∀ ∈ ∀ ∈ where gjk ,pk, ρ ≥0, pk is the transmission

power of radio k’s waveform, gkj is the link gain from the transmission source of radio k’s signal

to the point where radio j measures its interference, ρ(fk,fj) is the fraction of radio k’s signal

power that interferers with radio j. In general, ρ(fk,fj) is determined by the absolute value of the

correlation between the signal space basis functions parameterized by fk and fj. For signals with

flat power spectral densities and equal bandwidth B, ρ(fk,fj) can be expressed as in (3). For

signals where Fk represents an orthogonal set of channels, ρ(fk,fj) can be expressed as in (4).

( ) { }, max ,0 /j k j k

f f B f f Bρ = − − (3)

( ),

1

0

j k

j k

j k

f ff f

f fρ

==

≠ (4)

While the infinite loop DFS example failed to exhibit the BSI property4, for networks where a)

the BSI property does hold, b) each radio has the goal of minimizing (5), and c) each radio is

autonomously rational, all unilateral adaptations increase (6) as proven in Theorem 1.

( ) ( )\

,j kj k j k

k N j

I f g p f fρ∈

= ∑ (5)

( ) ( )1

1

,j

kj k j k

j N k

V f g p f fρ−

∈ =

= −∑∑ (6)

As (6) is just -Φ(f)/2, increases in V imply decreases in Φ so autonomously rational unilateral

adaptations guided by (5) must decrease Φ with each adaptation thereby satisfying the definition

of an IRN. In turn, this implies that networks formed from autonomously rational CRs guided by

(5) with BSI and unilateral timing do not have loops. This result is proven in Theorem 2.

4 Note that g31 = 1.0 and g13 = 0.5 where ρ is defined as in (4) and p3= p1 = 1.

9 of 26

Using the theory of potential games, [1] shows the conditions of BSI and exhaustive better

response dynamics guided by minimizing (5) are sufficient to guarantee convergence to a point f*

we call a unilateral minimizer (UM) such that there is no k N∈ such that

( ) ( )* * *, ,k k k kf f f f− −Φ < Φ� under unilateral timings and asynchronous timings. We say T is

asynchronous if for each t∈T each radio, k, has a probability ( )0,1kp ∈ of implementing its

decision rule.5 Thus the following set of conditions guarantee convergence of a DFS algorithm to

an UM of Φ(f).

C1. All CRs implement exhaustive better responses guided by (5).

C2. BSI holds.

C3. Radios adapt under unilateral or asynchronous timings.

All three of these conditions must be simultaneously satisfied for sufficiency. Note that the

infinite loop example satisfies C1 and C3 but not C2. An algorithm which is not exhaustive can

stop short of an UM of Φ(f). Under synchronous timing, consider the two channel, two link

network ( ) ( )1 2 1 2, ,F f f f f= × where g12 = g21, p1=p2, and ( )1 2,f fρ is given by (4). Under

synchronous timing (each radio adapts at each time in T), an exhaustive better response will

oscillate between ( )1 1,f f and ( )2 2,f f .

IV A DISTRIBUTED DFS ALGORITHM FOR INFRASTRUCTURE NETWORKS6

As implementing a random timer for decision timings and ensuring radios adapt to a channel

with less observed interference are both relatively straightforward tasks, the critical insight for

applying the IRN framework is finding a set of observations for which the BSI condition holds.

Now consider an 802.11a network with the 802.11h amendment for signaling between ANs

and clients. Suppose each AN maintains a table with |Fi| entries initialized to zero, corresponding

to the |Fi| channels available to the network. Whenever AN i detects an RTS/CTS or BSSID

5 For completeness, these theorems are proven without directly applying potential games in Theorems 3-6.

6 This algorithm was originally presented in our 2006 Milcom paper [Neel_06].

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signal from another AN, j, i stores the received power from j and the channel. While listening to

a channel, if multiple ANs transmit, measurements may or may not be possible. Also if the signal

power is too low for packet decoding, then no measurement can be made. When time for a

decision comes, i adds together the received powers from each observed AN to form an estimate

of the interference level in each channel and then applies an exhaustive better response rule to

the selection of an operating channel. This process is illustrated in Figure 1. Now let us make the

following assumptions about the network.

(A1) All RTS/CTS and BSSID messages are transmitted at the same power level – a reasonable

assumption as RTS/CTS messages are typically transmitted at maximum power to clear out

hidden nodes.

(A2) The ANs are not mobile so that the link budgets between ANs for each channel are

symmetric, i.e., gij = gji or that interference measurements are averaged over a sufficiently

long period so that gij = gji is approximately true.

(A3) Out of channel interference is negligible such that ( ),j kf fρ is given by (4).

(A1)-(A3) imply that ( ) ( ), ,ji j i j ij i j ig p f f g p f fσ σ= which means that BSI holds among the

ANs. Thus if we couple (A1-A3) with an exhaustive better response and ensure that the system

does not adapt synchronously, Theorem 2 assures us that the network will not become trapped in

an infinite loop and will converge to a minimizer of Φ(f).

The behavior of such a network is illustrated in a simulation of thirty ANs randomly

distributed over 1 km2 operating in an environment with a path loss exponent of 3 with random

placements and random initial channels and noise floors of -90 dBm. The radios are constrained

to operate in the eleven channels available in the 5.47-5.725 GHz European band (channels 100-

140) so the assumption A1 holds for all channels (in this case, signals are transmitted at 1 W).

The geographic distribution of the ANs and their final operating frequencies are shown in Figure

2 where a circle notes the position of an AN with its final channel id labeled just below and to

11 of 26

the right of the circle. Figure 3 depicts the transient behavior of the network with operational

channels for each AN (top), perceived interference levels by the ANs (middle), and the sum of

perceived interference levels (bottom) for the simulated network. Note that Φ(f) (bottom)

decreases with each adaptation thereby satisfying the definition of an IRN even though there are

instances of interference increasing for individual ANs (middle). Thus as is the case for all IRNs,

self-interested adaptations led to a socially desirable outcome (at least when socially desirable is

defined as the sum of observed network interference levels). As this algorithm converges to a

minimum of Φ(f), the algorithm performs at least as well as the centralized local search

algorithm for ANs in [2] if no restarts are employed. So, somewhat remarkably, this scalable

distributed low complexity algorithm yields results as good as a higher complexity centralized

algorithm – a rare case of a “free lunch” in an engineering application.

Performance of Client Devices

Of course, if the client devices are not also partaking of this free lunch, then there is little value

to the algorithm. Fortunately, the client devices also see a significant reduction in interference.

Figure 4 illustrates how the adaptations shown in Figure 3 impact the transient interference levels

measured by 50 client devices randomly scattered over the network with each client

communicating with its nearest AN. The top plot shows the frequencies of the client devices; the

middle plot shows the clients’ observed out-of-cluster interference levels7; and the bottom plot

shows the sum of the interference observed by the client devices. As part of modeling the out-of-

cluster interference observed by the clients, an infinite demand for data is assumed. With a

cluster with n clients, this assumption leads to a situation where each client is always contending

for access and thus is able to transmit only 1/nth

as frequently as the AN, which is involved in

every communication.

7 The proposed algorithm does not address intra-cluster interference and thus intra-cluster interference would not

change due to this algorithm.

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Unlike the interference observations made by the ANs, the sum interference for the client

observation statistics is not monotonically decreasing. This is because the BSI condition for this

algorithm only applies to the ANs. Fortunately, the assumption that client devices are associated

with its closest access point keeps the interference of clients and ANs reasonably correlated.

Policy Variations by Channel

If we permit the ANs to operate outside of channels 100-140, (A1) (pk = pi ∀i,k∈N) fails as the

lower and middle UNII bands (channels 36-64) limit transmission power levels to 200 mW [5]

instead of 1 W in the upper band. However, for non-overlapping signals,ρ(fi,fk)= ρ (fk,fi)=0, so

BSI still holds. Repeating the AN simulation to permit operation in channels 36-64 we get the

transient statistics shown in Figure 5 where it is evident that the network remains an IRN.

V AN IRN DFS ALGORITHM FOR AD-HOC NETWORKS

In an ad-hoc or peer-to-peer network, there is no topological justification for minimizing

interference at a particular node. If we wished to apply the infrastructure algorithm, we could

choose to have either end of each link act as the master node which minimizes its observed

interference and implement the preceding algorithms. However, the slave node would generally

have worse performance than the master node, though like the client devices in the preceding

section, we would expect better performance than if we did not implement the algorithm.

For this algorithm, we consider our decision making units as the links between nodes. To

apply the BSI condition under our assumption of equal transmit power and ρ given by (4), the

observed link gains between each pair of decision makers must be equal. Unlike our preceding

algorithms implemented on a single node, four different propagation paths exist between a pair

of links as illustrated in Figure 6. Thus while it is reasonable to assume that the link gain

between any two nodes is symmetric, gain symmetry between two links is less obvious.

Now let us arbitrarily label one side of a link k as ka and the other side as kb and the gain from

ka to mb as gkm(a,b). Then it is clear that when links k and m operate in the same band, gkm(a,b) =

13 of 26

gmk(b,a). Now let us define the following symbols to denote the sum, maximum, and minimum

gains from link k to link m.

• ( ) ( ) ( ) ( ), , , ,km km km km kmg g a a g a b g b a g b b= + + +�

(sum gain)

• ( ) ( ) ( ) ( ){ }ˆ max , , , , , , ,km km km km kmg g a a g a b g b a g b b= (maximum gain)

• ( ) ( ) ( ) ( ){ }min , , , , , , ,km km km km kmg g a a g a b g b a g b b=�

(minimum gain)

For fk = fm and by our assumption that gkm(a,b) = gmk(b,a), it is clear that kmg�

= mkg�

, ˆ ˆkm mkg g= ,

and km mkg g=� �

. Again each link k attempts to minimize its observed interference, but now the

link is minimizing observed interference as given by (7), (8), or (9) which are expressions for the

sum, maximum, and minimum interference that the other links induce to link k, respectively.

( ) ( ),k mk k m

m N

I f pg f fρ∈

= ∑� �

(7)

( ) ( )ˆ ˆ ,k mk k m

m N

I f pg f fρ∈

= −∑ (8)

( ) ( ),k mk k m

m N

I f pg f fρ∈

= −∑� � (9)

By applying the same reasoning used in Theorem 1 and Theorem 2, it is trivial to show that

reductions in (7), (8), and (9) lead to corresponding reductions in sum interference as measured

by (10), (11), and (12), respectively.

( ) ( )\

,kj k j k

j N k N j

f g p f fρ∈ ∈

Φ =∑ ∑� �

(10)

( ) ( )\

ˆ ˆ ,kj k j k

j N k N j

f g p f fρ∈ ∈

Φ =∑ ∑ (11)

( ) ( )\

,kj k j k

j N k N j

f g p f fρ∈ ∈

Φ =∑ ∑� � (12)

When CRs implement exhaustive better responses to minimize (7), (8), or (9), the system will

converge to a minimizer of (10), (11), and (12), respectively where (10) is an expression for the

sum interference. Accordingly, we focus on implementations of (7) in the following.

14 of 26

To implement such a system, the algorithm of Figure 1 could be modified as shown in Figure

7. First, the algorithm is implemented on each node in the network, but during the link

establishment phase one node in the link is designated as ‘a’ and the other as ‘b’. When a

RTS/CTS message is detected, the node notes 1) both addresses contained in the message to

ascertain the link being observed, 2) the address of the transmitting node, 3) the received signal

power, 4) the channel being listened to, Lc. The node then updates the corresponding entry in its

interference table. When it is time for a decision (triggered on either the a or b node), the a and b

nodes combine their interference tables to form (7), (8), or (9). Node a then applies an

exhaustive better response to select a channel that reduces (7), (8), or (9) depending on the

implementation and then signals the change to node b.

The transient operation of such a network with 30 links minimizing (7) with random initial

frequencies and locations, random link distances of 36m or less, and a log-normal path loss

model (n=3, σ=10dB) is illustrated the simulation results shown in Figure 8. These results are

similar to the infrastructure network because both systems conform to the same game model and

are thus predicted to behave in a similar manner.

VI AGGREGATE STATISTICS

Although theory guarantees convergence to an UM of Φ, it does not specify the improvement

gain this system would experience as such gains are highly dependent on the initial configuration

of the ANs and their relative locations. For insight into the performance gain realized by this

system, we conducted repeated simulations of varying numbers of 802.11a ANs randomly

distributed over 1 km2 with random initial frequencies chosen from the 18 European UNII

channels. This simulation was conducted for 5, 10, 15, 18, 20, 25, 30, 35, 40, 50, 60, 70, 80, and

100 ANs with 500 random trials for each number of ANs. The results of this simulation are

presented in Figure 9 where each circle depicts the reduction in Φ(f) for a single trial, and the

line traces out the average reduction in Φ(f). As can be seen for AN densities > 40/km2

the

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typical reduction in sum network interference was about 19 dB over the system’s initial random

frequency assignment with less improvement seen for lower AN densities.

Figure 10 shows a similar simulation with twice as many client devices as ANs but with only

100 random trials per number of ANs and assuming an infinite data demand model. Comparing

the aggregate statistics, it is seen that the algorithm yields a greater reduction in interference for

the client devices than for the ANs’ observed interference for low density deployments with the

situation reversed for high density deployments. In all cases, however, the ANs’ actual

interference is reduced by more than the clients’ interference and more than the reduction in

interference observed by the ANs. This phenomenon is due to two factors. First, the AN

observations only include the interference from other ANs. In comparison, the client interference

includes interference from both other client devices and from ANs. Thus a greater possible

reduction in interference is possible for the client interference statistics. Second, for 18 or fewer

ANs (36 or fewer client devices), frequency allocations where all clusters are operating on a

unique channel are the only NE for the network. Thus no cell-edge client-to-client interference

effects would be expected at the steady-state frequency allocation for these scenarios. However,

for greater numbers of ANs, this cell-edge effect becomes non-negligible, particularly for the

client devices. So the gain experienced is reduced, but the average improvement seen by

adapting away from a random frequency reuse pattern remains significant – greater than 11 dB.

We performed a similar sequence of experiments for the ad-hoc algorithm links uniformly

distributed over a 0.5 x 0.5 km area and random initial frequencies with simulation parameters as

before. Figure 11 shows the average interference levels before and after application of the

algorithm and the worst performing link averaged over the simulation runs for each number of

links. Until the number of decision makers exceeds the number of available channels (18), the

final interference levels remain at the noise floor (-90 dBm) in line with Theorem 6. The average

interference level is reduced by approximately 9 dB and the average of the worst performing

links for each simulation is reduced by approximately 30 dB. Thus the most significant benefit

16 of 26

from this algorithm is the improvement in worst-case performance. Note that this improvement

in worst case performance also captures a significant fraction of the improvement in Φ(f) as Φ(f)

tends to be dominated by its largest term.

This reduction in interference leads to a significant increase in the number of links whose

interference levels fall below the collision threshold. Section 17.3.10.5 of [17] specifies that the

collision avoidance algorithms must be capable of detecting signals received at a signal power of

-82 dBm with 90% accuracy with signals detected below this threshold presumably ignored.8 By

applying this threshold to the preceding simulation and assuming 100% detection accuracy, we

formed the graph shown in Figure 11. On average with 100 links operating in the 0.5 x 0.5 km2

area, 75% of the links operate collision free while just 25% of the links were operating collision

free without the algorithm.

VII CONCLUSIONS

As we showed in the introduction, even relatively simple dynamic spectrum access algorithms

can have significant drawbacks when implemented in a network. Previously, this problem has

been mitigated by adding significant signaling and complexity to the network to limit the impact

of interactive decision processes. By applying the game theory inspired IRN framework of [1],

this paper proposed low complexity autonomous distributed infrastructure and ad-hoc DFS

algorithms whose adaptations looplessly converge to a unilateral minimizer of the sum of

observed interference levels without coordination or information exchange between decision

makers. We showed that these non-cooperative non-collaborative algorithms can be incorporated

into systems where different power policies apply to different orthogonal channels or different

radios. The algorithms were shown to provide the greatest benefit to the most interfered with

links and yielded an average reduction in sum network interference of 19 dB, average link

interference of 9 dB, and the interference of the most interfered with link by 30 dB. It was seen

that while the infrastructure-based algorithm privileges the performance of the AN over the

8 In practice, how a device treats lower power signals is vendor specific.

17 of 26

clients, the clients also experienced a significant reduction in interference. It was empirically

shown that this reduction in interference results in a significant increase in the percentage of

links which operate free of collisions. Future publications will consider how the algorithms can

be modified to accommodate noisy estimations, in-channel power variations, frequency sets

which vary by radio, and the presence of legacy devices. Additionally, it will be shown in later

publications how these algorithms can be combined with ad-hoc routing protocols, bandwidth

allocation, and power control to ensure distributed autonomously rational cognitive radios

looplessly converge to desirable operating states.

VIII REFERENCES

[1] J. Neel, R. Menon, A. MacKenzie, J. Reed, R. Gilles, “Interference Reducing Networks,”

Submitted to ACM/Springer Mobile Networks and Applications (MONET) Special Issue on

“Cognitive Radio Oriented Wireless Networks and Communications”. (Draft available at

www.mprg.org/gametheory)

[2] K. Leung, B. Kim, “Frequency Assignment for 802.11n Wireless Networks,” VTC 2003, vol.

3, pp. 1422-1426, Oct. 6-9 2003.

[3] M. Steenstrup, “Opportunistic use of radio-frequency spectrum: a network perspective,”

DySPAN2005, Nov. 2005 pp. 638-641.

[4] J. Zhao, H. Zheng, G. Yang, “Distributed Coordination in Dynamic Spectrum Allocation

Networks,” DySPAN 2005, pp. 269-278, Nov. 2005.

[5] R. Etkin, A. Parekh, D. Tse, “Spectrum Sharing for Unlicensed Bands,” DySPAN2005, pp.

251-258, Nov. 2005.

[6] N. Nie, C. Comaniciu, “Adaptive channel allocation spectrum etiquette for cognitive radio

networks,” DySPAN2005, Nov. 2005 pp. 269-278.

[7] C. Sung, K. Leung , “On the stability of distributed sequence adaptation for cellular

asynchronous DS-CDMA systems,” IEEE Transactions on Information Theory, vol. 49, no.

7, July 2003, pp. 1828-1831.

18 of 26

[8] Y. Xing, R. Chandramouli, S. Mangold, S. Shankar, “Dynamic spectrum access in open

spectrum wireless networks,” IEEE Journal on Selected Areas in Communications, vol. 24, No. 3,

pp. 626-637, March 2006.

[9] E.Garcia Villegas, R. Vidal Ferro, J. Paradells Aspas. “Implementation of a Distributed

Dynamic Channel Assignment Mechanism for IEEE 802.11 Networks,” 16th International

Symposium on Personal, Indoor and Mobile Radio Communications, pp. 1458-1462, 2005.

[10] C. Sung, K.W. Shum and K. Leung, “Multi-objective power control and signature sequence

adaptation for synchronous CDMA systems - a game-theoretic viewpoint”, Proceedings of the

IEEE International Symposium on Information Theory, July 2003, pp. 335-335.

[11] J. Hicks, A. MacKenzie, J. Neel, J. Reed, “A Game Theory Perspective on Interference

Avoidance,” IEEE GlobeCom, vol.1, pp. 257-261, Dec. 2004.

[12] R. Menon, A. MacKenzie, R. Buehrer, J. Reed, “Game Theory and Interference Avoidance

in Decentralized Networks” SDR Forum Technical Conference Nov. 15-18, 2004.

[13] S. Ulukus and R.D. Yates, “Iterative construction of optimum signature sequence sets in

synchronous CDMA systems,” IEEE Transactions on Information Theory, vol. 47, no. 5, July

2001, pp. 1989-1998.

[14] H. Luo, N. Shankaranarayanan, “A distributed dynamic channel allocation technique for

throughput improvement in a dense WLAN environment,” ICASSP ’04, vol. 5, pp. 345-348, May

17-21, 2004.

[15] J. Neel, R. Buehrer, J. Reed, and R. Gilles, “Game Theoretic Analysis of a Network of

Cognitive Radios,” Midwest Symposium on Circuits and Systems 2002, pp. III-409-III-412.

[16] J. Neel. J. Reed, A. MacKenzie, “Cognitive Radio Network Performance Analysis,” in

Cognitive Radio Technology, B. Fette, ed., Elsevier August, 2006.

[17] IEEE Std 802.11a-1999(R2003), Reaffirmed 12 June 2003.

[Neel_06] J. Neel, J. Reed, “Performance of Distributed Dynamic Frequency Selection Schemes

for Interference Reducing Networks,” Milcom 2006. Washington D.C. Oct 23-25, 2006.

19 of 26

APPENDIX A

We also showed the following theorems in [1], but via the use of potential games. Without

introducing potential games, we prove the same results under conditions C1-C3.

Theorem 1:9 Each adaptation increases (6).

Proof: Note that ( ) ( ) ( ) ( )1 2 1 2

\ \

, , , ,j j j j j j kj k j k kj k j k

k N j k N j

I f f I f f g p f f g p f fρ ρ− −∈ ∈

− = −∑ ∑ and

( ) ( ) ( ) ( )1 2 1 2

\ \

, , , ,j j j j kj k j k kj k j k

k N j k N j

V f f V f f g p f f g p f fρ ρ− −∈ ∈

− = − +∑ ∑ . A unilateral

autonomously rational adaptation from ( )1,j j

f f− to ( )2 ,j j

f f− by radio j implies

( ) ( )1 2, ,j j j j j j

I f f I f f− −> . The preceding relationships then imply ( ) ( )1 2, ,j j j j j j

V f f V f f− −< . ■

Theorem 2: If Φ is monotonically decreasing for a sequence of frequency vectors, ( )1:

k

k nf

=,

then there are no ( )1:

, k

m nk n

f f f=

∈ such that m nf f= with m n≠ .

Proof: Suppose ( )1:

, k

m nk n

f f f=

∈ , m nf f= , m n≠ , and ( ) ( ) ( )1

1:

k k k k

k nf f f f

+

=Φ > Φ ∀ ∈ . Then

we have ( ) ( )m nf fΦ > > Φ� , but this contradicts the assumption that m nf f= .■

Theorem 3: If F is finite, then all sequences of unilateral exhaustive better responses are finite.

Proof: For finite F, infinite sequences imply existence of a loop. By Theorem 2, no loops exist. ■

Theorem 4: The vector f* is a fixed point of 1 :nd d d F F= × × →� iff f

* is an UM of Φ.

Proof: Suppose f* is fixed point of d but not an UM of Φ. Then k kf F∃ ∈� such that

( ) ( )* *, ,k k k kI f f I f f− −<� contradicting C1. Suppose f* is an UM of Φ and not a fixed point of d.

Then k kf F∃ ∈� such that ( ) ( )* *, ,k k k kI f f I f f− −<� which implies ( ) ( )* *, ,k k k kf f f f− −Φ < Φ� .■

9 This result is shown in [1] by demonstrating that the BSI condition implies that the network with radios guided by

the goal uj=-Ij forms an exact potential game with exact potential function (6). Though beyond the scope of this

paper, a longer discussion of potential games is given in [1] and [16].

20 of 26

Theorem 5: Let MΓ be the Markov chain that results when players in Γ=⟨N,F,{ui}⟩ operate under

asynchronous timing. For finite F, MΓ is an absorbing Markov chain.

Proof: A Markov chain is absorbing if from every state there exists a sequence of state

transitions of nonzero probability that terminates in an absorbing state. By Theorem 3, from

every initial 1f F∈ , there exists a sequence of unilateral adaptations that ends in a fixed point of

d and every sufficiently long sequence of adaptations (no longer than |F|) arrives at a fixed point

of d. Further, it is apparent that every fixed point for d is also an absorbing state for MΓ. For each

f F∈ , let γf be a sequence that terminates in an absorbing state for unilateral timing and

( ),fn kγ be the player that adapts at step k of γf and n(γf) be the sequence of adapting players in γf.

Under an asynchronous timing model, the probability that only n(γf) adapts at a particular

iteration is given by ( ) ( )

( ),

\ ,

1f

f

mn km N n k

p pγ

γ∈

−∏ and that starting from f, the probability the exact

sequence of adapting players is n(γf) is given by ( ) ( )

( )( ) ( ) ,

,\,

1f

f f n kf

mn km Nn k n

p p

γ

γγ γ ∈∈

−

∏ ∏ . As this last

expression is greater than zero and as every fixed point for d is an absorbing state of MΓ, γf

specifies a sequence of state transitions of nonzero probability that terminates in an absorbing

state. ■

Theorem 6: If kN F k N≤ ∀ ∈ and ρ is given by (4), then f* is a fixed point of d iff Ik(f) = 0

k N∀ ∈ .

Proof: Suppose f* is a fixed point of d, but Ik(f) ≠ 0 for some k∈N. Then two CRs operate on the

same channel. However, as kN F k N≤ ∀ ∈ , there must exist an unoccupied channel which k

can switch too. Therefore, f* is not a fixed point of d. Suppose Ik(f

*) = 0 k N∀ ∈ and f is not a

fixed point. Thus there must be some adaptation such that Ik(f) < 0. But with g, p, ρ ≥0, Ik(f) can

never be less than 0. Therefore f* must be a fixed point. ■

21 of 26

FIGURES

Figure 1: Flowchart of Algorithm Implemented on Each AN.

Figure 2: Steady-state Channels Selected for a Random Distribution of ANs with Random Initial

Channels in the 5.47-5.725 GHz Band

Listen on

Channel LC

RTS/CTS

energy detected?Measure power of access node in message, p

Note address of access

node, a

Update interference

tableTime for decision?

Apply decision criteria for new

operating

channel, OCUse 802.11h to signal change

in OC to clients

yn

Pick channel to

listen on, LC

y

n

Start

22 of 26

Figure 3: Transient Statistics for the Network in Figure 2

Figure 4: Transient Statistics for 50 Client Devices for the Network in Figure 2

23 of 26

Figure 5: Transient Statistics with Policy Variations by Channel

1a

1b

2a

2b

g12(a,a)= g21(a,a)

g12(b,b) = g21(b,b)

g 12(b

,a)

g12(a,b)

g21 (b,a)

g 21(a

,b)1a

1b

2a

2b

g12(a,a)= g21(a,a)

g12(b,b) = g21(b,b)

g 12(b

,a)

g12(a,b)

g21 (b,a)

g 21(a

,b)

Figure 6: Propagation Paths (solid) between Links 1 and 2 (dashed)

24 of 26

Listen on

Channel LC

RTS/CTS

energy detected?Measure power

of node

in message, p

Note addresses of nodes from

RTS/CTS

Update

interference table

Time for decision?

Apply decision

criteria for new operating

channel, OC

Signal change

If needed

yn

Pick channel to

listen on, LC

y

n

StartMap to link

If ‘b’ node, transmit

relevant interference levels

to ‘a’ node

Listen on

Channel LC

RTS/CTS

energy detected?Measure power

of node

in message, p

Note addresses of nodes from

RTS/CTS

Update

interference table

Time for decision?

Apply decision

criteria for new operating

channel, OC

Signal change

If needed

yn

Pick channel to

listen on, LC

y

n

StartMap to link

If ‘b’ node, transmit

relevant interference levels

to ‘a’ node

Figure 7: Algorithm for Implementing DFS for Ad-Hoc Network

0 50 100 150 200 250 300

40

60

80

100

120

140

Channel

0 50 100 150 200 250 300-90

-80

-70

-60

I i(f)

(dB

m)

0 50 100 150 200 250 300-80

-75

-70

-65

-60

-55

iteration

Φ(f

) (d

Bm

)

Figure 8: Transient Statistics for a Simulation of 30 Randomly Distributed Links

25 of 26

Figure 9: Aggregate Steady-state Statistics for Observed AN Interference

Figure 10: Comparison of Reductions in Aggregate Interference for Infrastructure Algorithm

0 10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60

70

Number of Access Nodes

Reduction in N

et

Inte

rfere

nce (

dB

)

26 of 26

0 10 20 30 40 50 60 70 80 90 100-90

-85

-80

-75

-70

-65

-60

-55

-50

-45

Ste

ady-s

tate

Inte

rfere

nce levels

(dB

m)

Number Links

Typical Worst Case Without Algorithm

Average Without Algorithm

Typical Worst Case With Algorithm

Average With Algorithm

Colission Threshold

Figure 11: The Ad-hoc Algorithm Reduces Average Interference Levels by 9 dB and the Interference of

the Worst Performing Links by 30 dB

0 10 20 30 40 50 60 70 80 90 10010

20

30

40

50

60

70

80

90

100

Perc

enta

ge o

f C

olli

sio

n F

ree L

inks

Number of Links

Worst Case Without Algorithm

Average Without Algorithm

Worst Case With Algorithm

Average With Algorithm

Figure 12: The Algorithm Significantly Reduces the Number of Potential Collisions